Kent Distribution Research Articles

Goodness-of-fit for a concentrated von Mises-Fisher distribution

The von Mises-Fisher distribution is widely used for modelling directional data. In this paper we propose goodness-of-fit methods for a concentrated von Mises-Fisher distribution and we analyse by simulation some questions concerning the application of these tests. We analyse the empirical power of the Kolmogorov-Smirnov test for several dimensions of the sphere, supposing as alternative hypothesis a mixture of two von Mises-Fisher distributions with known parameters. We also compare the empirical power of the Kolmogorov-Smirnov test with the Rao’s score test for data on the sphere, supposing as alternative hypothesis, a mixture of two Fisher distributions with unknown parameters replaced by their maximum likelihood estimates or a 5-parameter Fisher-Bingham distribution. Finally, we give an example with real spherical data.

Mocapy++ - A toolkit for inference and learning in dynamic Bayesian networks

BackgroundMocapy++ is a toolkit for parameter learning and inference in dynamic Bayesian networks (DBNs). It supports a wide range of DBN architectures and probability distributions, including distributions from directional statistics (the statistics of angles, directions and orientations).ResultsThe program package is freely available under the GNU General Public Licence (GPL) from SourceForge http://sourceforge.net/projects/mocapy. The package contains the source for building the Mocapy++ library, several usage examples and the user manual.ConclusionsMocapy++ is especially suitable for constructing probabilistic models of biomolecular structure, due to its support for directional statistics. In particular, it supports the Kent distribution on the sphere and the bivariate von Mises distribution on the torus. These distributions have proven useful to formulate probabilistic models of protein and RNA structure in atomic detail.

On the Fisher–Bingham distribution

This paper primarily is concerned with the sampling of the Fisher---Bingham distribution and we describe a slice sampling algorithm for doing this. A by-product of this task gave us an infinite mixture representation of the Fisher---Bingham distribution; the mixing distributions being based on the Dirichlet distribution. Finite numerical approximations are considered and a sampling algorithm based on a finite mixture approximation is compared with the slice sampling algorithm.

Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants

The Fisher–Bingham distribution is obtained when a multivariate normal random vector is conditioned to have unit length. Its normalising constant can be expressed as an elementary function multiplied by the density, evaluated at 1, of a linear combination of independent noncentral χ12 random variables. Hence we may approximate the normalising constant by applying a saddlepoint approximation to this density. Three such approximations, implementation of each of which is straightforward, are investigated: the first-order saddlepoint density approximation, the second-order saddlepoint density approximation and a variant of the second-order approximation which has proved slightly more accurate than the other two. The numerical and theoretical results we present showthat this approach provides highly accurate approximations in a broad spectrum of cases.

The simulation of spherical distributions in the Fisher-Bingham family

The problem of generating pseudo-random unit vectors from the Fisher-Bingham distribution is considered. Attention is focussed on a subfamily known as FB6' which includes many of the spherical distributions of practical interest, including those of Dimroth-Watson, FB4' FB5 and Bingham type. The rejection procedures suggested here for these distributions should prove adequate for most practical purposes. Possibilities for simulating from the general Fisher-Bingham distribution are also mentioned briefly.